The Proper and Semi-proper Forcing Axioms for Forcing Notions That Preserve

We prove that the PFA lottery preparation of a strongly unfoldable cardinal κ under ¬0 forces PFA(א2-preserving), PFA(א3-preserving) and PFAא2 , with 2 ω = κ = א2. The method adapts to semi-proper forcing, giving SPFA(א2-preserving), SPFA(א3-preserving) and SPFAא2 from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction SPFA(א2-preserving)+ SPFA(א3-preserving)+SPFAא2 +2ω = א2. Since unfoldable cardinals are relatively weak as large cardinal notions, our summary conclusion is that in order to extract significant strength from PFA or SPFA, one must collapse א3 to א1. A convergence of recent results has pointed at the strongly unfoldable cardinals, relatively low in the large cardinal hierarchy, as a surprisingly efficacious substitute for supercompact cardinals in several large cardinal phenomena. For example, Johnstone [Joh07, Joh], fulfilling earlier hints that a Laver-style indestructibility phenomenon may be possible, proved that any strongly unfoldable cardinal κ can be made indestructible by all <κ-closed κ-proper forcing. In subsequent joint work [HJ], we extended this to indestructibility by all <κ-closed κ-preserving forcing. In this article, we adapt the construction, just as Baumgartner modified the Laver preparation [Lav78] to force the Proper Forcing Axiom (PFA) from a supercompact cardinal, to force interesting fragments of PFA from an unfoldable cardinal. Succinctly, in this article we explain how to carry out the Baumgartner PFA construction using a mere strongly unfoldable cardinal in the place of his supercompact cardinal and using the PFA lottery preparation in place of his Laver-style iteration. The end result is a new relatively low upper bound on PFA(א2-preserving) and PFA(א3-preserving). The methods extend analogously to the case of semi-proper forcing and the corresponding fragments of SPFA. This project can be viewed as a continuation of Miyamoto’s construction [Miy98], where he used a hypothesis equivalent to strong unfoldability to prove the relative consistency of PFAc (see Theorem 6 below). There is also a strong affinity Received by the editors November 20, 2007, and, in revised form, August 13, 2008. 2000 Mathematics Subject Classification. Primary 03E55, 03E40.